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Unformatted text preview: Jaime Frade ECO5282Dr. Garriga Homework #4 1. (Equilibrium with Exogenous Borrowing Constraints) Consider an economy with a continuum [0,1] of consumers of symmetric typed who live forever. Consumers have utility X t =0 t log( c i t ) Consumers of type 1 have an endowment steam of the single good in each period ( w 1 ,w 1 1 ,w 1 2 ,... ) = (8 , 1 , 8 , 1 ,... ) while consumers of type 2 have ( 2 , 2 1 , 2 2 ,... ) = (1 , 8 , 1 , 8 ,... ). In addition there is one unit of trees that produses d = 1 units of good at each period. Each consumer of type i owns s i of such a trees in period t = 0, s i &gt; 0, s 1 + s 2 = 1. Trees do not grow or decay over time. (a) (i) Define an ArrowDebreu equilibrium. Definition An ArrowDebreu equilibrium is an allocation {{ c i , s i t +1 } t =0 } 2 i =1 and a sequence of prices { q t ,r t } t =0 such that the allocation solves each household problem. For a given household, the com sumer solves max X t =0 t u ( c i t ) (1) Subject to the following budget constraint c i t + q t ( s i t +1 s i t ) i t + d s i t t (2) s i t Goods and financial markets clear c 1 t + c 2 t = g + b + d = t (3) s 1 t + s 2 t 1 t (4) From the initial problem, a symmetric allocation implies the following: c * = b c 1 t = b c 2 t = g + b + d 2 = 8 + 1 + 1 2 = 5 t (5) (ii) Compute the equilibrium prices. The Euler equation for a symmetric equilibrium is also satisfied. Since u ( c g ) = u ( c b ) and u ( c g ) = u ( c b ) obtain the following: u ( c g ) u ( c b ) = q + d q = u ( c g ) u ( c b ) (6) 1 = q + d q Then, using d = 1, the equilibrium prices satisfies: q = 1 d = 1 (7) 1 (iii) Given the endowment, find the intial asset holdings s 1 and s 2 such that in equilibrium, b c 1 t = b c 2 t t . [ Hint: You have to becareful when writing down the consolidated budget constraint for the individuals.] When the financial markets clear, (14), then, the aggregate resource constraint as well as the consumer budget constraint are satisfied. Compute the steady state trade associated to the optimal consumption level: 2 g = ( p + d ) s 2 p s 1 b 2 = ( p + d ) s 1 p s 2 Solve for the optimal share distribution by solving a linear system of equations: p + d p p p + d s 2 s 1 = 2 g b 2 (8) Consumers of type 1 will recieve the a good shock in the intial period, while consumers of type 2 recieve a bad shock. Compute the intial asset holdings s 1 and s 2 , using the following subsitutions for the equilibrium price (7), d = 1, g = 8, b = 1, and 2 = 5 into (8), to obtain: p + d p p p + d s 2 s 1 = 2 g b 2 1 1  1  1 1 1 s 2 s 1 = 3 4 Solving for s 1 and s 2 in the previous equation will obtain the following:...
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